Dans cette thèse on étudie les effets de taille finie au-dessus de la dimension critique supérieure d_c. Les effets de taille finie y ont longtemps été incomplètement compris, en particulier vis-à-vis de leur dépendance en fonction des conditions aux limites. La violation de la relation d’échelle dite d’hyperscaling a été l’un des aspects les plus évidents des difficultés rencontrées. Le désaccord avec le scaling usuel est dû au caractère de variable non pertinente dangereuse du terme de self-interaction dans la théorie en ϕ^4. Celle-ci était considérée comme dangereuse pour la densité d’énergie libre et les fonctions thermodynamiques associées, mais pas dans le secteur des corrélations. Récemment, un schéma nouveau de scaling a été proposé dans lequel la longueur de corrélation joue un rôle central et est également affectée par la variable non pertinente dangereuse. Ce nouveau schéma, appelé QFSS, est basé sur le fait que la longueur de corrélation exhibe au lieu du scaling usuel ξ~L un comportement en puissance de la taille finie ξ~L^ϙ. Ce pseudo-exposant critique ϙ est lié à la dimension critique supérieure et à la variable dangereuse. Au-dessous de d_c, cet exposant prend la valeur ϙ=1, mais au-dessus, il vaut ϙ=d/d_c. Le schéma QFSS est parvenu à réconcilier les exposants de champs moyen et le Finite-Size-Scaling tel que dérivé du Groupe de Renormalisation pour les modèles avec interactions à courte portée au-dessus de d_c en conditions aux limites périodiques. Si ϙ est un exposant universel, la validité de la théorie doit toutefois s’étendre également aux conditions de bords libres. Des tests initiaux dans de telles conditions ont mis en évidence de nouvelles difficultés: alors que le QFSS est valable au point pseudo-critique auquel les grandeurs thermodynamiques telles que la susceptibilité manifestent un pic à taille finie, au point critique on a pensé que c’était le FSS standard qui prévalait avec les exposants de champ moyen et ξ~L. On montre dans ce travail qu’il en va différemment de la situation au point critique et qu’à la place ce sont les exposants gaussiens qui s’appliquent en l’absence de variable non pertinente dangereuse. Pour mettre en évidence ce résultat, nous avons mené des simulations de modèles avec interactions à longue portée, qui peuvent être à volonté étudiés au-dessus de leur dimension critique supérieure. Nous avons aussi développé une étude des modes de Fourier qui permet de fournir des exemples de quantités non affectées par la présence de la variable non pertinente dangereuse / In this project finite-size size scaling above the upper critical dimension〖 d〗_c is investigated. Finite-size scaling there has long been poorly understood, especially its dependency on boundary conditions. The violation of the hyperscaling relation above d_c has also been one of the most visible issues. The breakdown in standard scaling is due to the dangerous irrelevant variables presented in the self-interacting term in the ϕ^4 theory, which were considered dangerous to the free energy density and associated thermodynamic functions, but not to the correlation sector. Recently, a modified finite-size scaling scheme has been proposed, which considers that the correlation length actually plays a pivotal role and is affected by dangerous variables too. This new scheme, named QFSS, considers that the correlation length, instead of having standard scaling behaviour ξ~L , scales as ξ~L^ϙ. This pseudocritical exponent is connected to the critical dimension and dangerous variables. Below d_c this exponent takes the value ϙ=1, but above the upper critical dimension it is ϙ=d/d_c. QFSS succeeded in reconciling the mean-field exponents and FSS derived from the renormalisation-group for the models with short-range interactions above d_c with periodic boundary conditions. If ϙ is an universal exponent, the validity of that theory should also hold for the free boundary conditions. Initial tests for such systems faced new problems. Whereas QFSS is valid at pseudocritical points where quantities such as the magnetic susceptibility experience a peak for finite systems, at critical points the standard FSS seemed to prevail, i.e., mean-field exponents with ξ~L. Here, we show that this last picture at critical point is not correct and instead the exponents that applied there actually arise from the Gaussian fixed-point FSS where the dangerous variables are suppressed. To achieve this aim, we study Ising models with long-range interaction, which can be tuned above〖 d〗_c, with periodic and free boundary conditions. We also include a study of the Fourier modes which can be used as an example of scaling quantities without dangerous variables

Dans cette thèse, nous présentons des études théoriques de matériaux pérovskites sous con-trainte mécanique uniaxiale en combinant les calculs de premier principe DFT ainsi quela théorie phénoménologique de type Landau. Les pérovskites ABO3 forment une classetrès importante de matériaux fonctionnels, qui peuvent présenter un large éventail de pro-priétés (e.g., supraconductivité, magnétisme, ferroélectricité, multiferroïcité, transitionsmétal-isolant. . . ) grâce aux petites distorsions d’ une même structure prototype cubique.Bien que ces composés aient été largement étudiés expérimentalement et théoriquement, ilreste encore des questions importantes et non résolues concernant les effets de contraintesuniaxiales. Au cours de ces dernières années, l’ ingénierie de contrainte a été décrite commeune approche originale pour ajuster les propriétés ferroélectriques pérovskites ABO3. Alorsque les effets de tension épitaxié-biaxiale et pression la hydrostatique, sont plutôt bien com-pris dans cette classe de matériaux, très peu est connu en ce qui concerne l’ effet des con-traintes mécaniques uniaxiales. Notre étude est motivée par ce manque de compréhensionactuelle de l’ effet de tension et compression uniaxiale, qui a été jusqu’à présent presquetotalement négligé. Deux composés prototypes sont étudiés dans le détail: PbTiO3 etSrTiO3. Après une introduction générale sur les composés ABO3 et les calculs techniques(ab initio et modèle phénoménologique de Landau), nous avons étudié l’ effet de contraintesmécaniques sur ces matériaux dans notre thèse.PbTiO3 est un composé ferroélectrique prototypique et également l’ un des composantsmère de la solution solide Pb(Zr,Ti)O3 (PZT), qui est le piézoélectrique le plus largementutilisé dans des applications. Pour PbTiO3, nous avons montré que indépendammentde la contrainte mécanique uniaxiale appliquée, le système conserve un état fondamentalpurement ferroélectrique avec la polarisation alignée, soit le long de la direction de lacontrainte (en phase FEz) ou bien le long d’ un des axes pseudo-cubique, qui lui estperpendiculaire (phase de FEx). Cela contraste avec les cas de contraintes mécaniquesisotropes ou bi-axial, pour qui de nouvelles phases combinant des modes ferroélectriqueset antiferrodistortives ont déjà été décrites. Sous contrainte uniaxiale, PbTiO3 passe d’unétat fondamental FEx sous compression à un état fondamental FEz en tension au-delà d’une tension critique !czz! +1%. Sous contrainte uniaxiale, PbTiO3 présente soit un étatfondamental FEx sous compression ("zz < 0) ou un état fondamental de FEz sous tension("zz > 0). Cependant, ici, un brusque saut des paramètres structuraux est prévu sousdes contraintes de compression et de traction à des valeurs critiques "zz! +2 GPa et −8GPa. Ce comportement semble similaire à celui pré-prédit sous pression isotrope négativeet pourrait se révéler utile en pratique pour améliorer la réponse piézoélectrique dans lesnano-composants.Le deuxième composé intéressant est SrTiO3. Il a été largement étudié au cours desdernières décennies, en raison de ses propriétés exceptionnelles à basse température. Dansce travail, nous avons élargi nos précédentes études de PbTiO3, en explorant théorique-ment les effets de pression sur la perovskite SrTiO3, combinant les premiers principes decalculs et un modèle phénoménologique de type Landau. Nous avons discuté de l’évolutiondes fréquences des phonons de SrTiO3 des trois cas de contraintes isotrope, uniaxial ettensions biaxiaux en utilisant les calculs de premier principe. Nous confirmons des travauxexpérimentaux précédents sur SrTiO3 que ça soit en contrainte épitaxiée ou sous pressionhydrostatique. Enfin, nous avons calculé de diagramme de phase de SrTiO3 sous contrainteuniaxiale, obtenue à partir de la théorie de Landau que nous avons comparé aux calculsde premier principe. / In the present thesis we present theoretical studies of perovskite compounds under uniax-ial mechanical constraints combining first-principles DFT calculations approach and phe-nomenological Landau theory. ABO3 perovskites form a very important class of functionalmaterials that can exhibit a broad range of properties (e.g., superconductivity, magnetism,ferroelectricity, multiferroism, metal-insulator transitions. . . ) within small distortions ofthe same simple prototype cubic structure. Though these compounds have been exten-sively studied both experimentally and computationally, there are still unresolved issuesregarding the effect of pressure. In recent years, strain engineering has reported to bean original approach to tune the ferroelectric properties of perovskite ABO3 compounds.While the effect of epitaxial biaxial strain and hydrostatic strain is rather well understoodin this class of materials, very little is yet known regarding the effect of uniaxial mechanicalconstraints. Our study is motivated by the little existing understanding of the effect ofuniaxial strain and stress, that has been up to now almost totally neglected. Two proto-type compounds are studied in detail: PbTiO3 and SrTiO3. After a general introductionon ABO3 compounds and calculations techniques (ab initio and phenomenological Landaumodel), we studied the effect of mechanical constraints in these compounds in our thesis.PbTiO3 is a prototypical ferroelectric compound and also one of the parent components ofthe Pb(Zr,Ti)O3 solid solution (PZT), which is the most widely used piezoelectrics. ForPbTiO3, we have shown that irrespectively of the uniaxial mechanical constraint applied,the system keeps a purely ferroelectric ground-state, with the polarization aligned eitheralong the constraint direction (FEz phase) or along one of the pseudocubic axis perpen-dicular to it (FEx phase). This contrasts with the case of isotropic or biaxial mechanicalconstraints for which novel phases combining ferroelectric and antiferrodistortive motionshave been previously reported. Under uniaxial strain, PbTiO3 switches from a FEx groundstate under compressive strain to FEz ground-state under tensile strain, beyond a critical strain !czz! +1%. Under uniaxial stress, PbTiO3 exhibits either a FEx ground state undercompression ("zz < 0) or a FEz ground state under tension ("zz > 0). Here, however, anabrupt jump of the structural parameters is also predicted under both compressive andtensile stresses at critical values "zz! +2 GPa and −8 GPa. This behavior appears similarto that predicted under negative isotropic pressure and might reveal practically useful toenhance the piezoelectric response in nanodevices.The second compound of interest is SrTiO3. It has been widely studied in the past decadesdue to its unusual properties at low temperature. In this work, we have extended ourprevious investigations on PbTiO3 by exploring theoretically the pressure effects on per-ovskite SrTiO3 combining the first-principles calculations and a phenomenological Landaumodel. We have discussed the evolution of phonon frequencies of SrTiO3 with the threeisotropic, uniaxial and biaxial strains using first-principles calculations. We also reproducethe previous work done in SrTiO3 with epitaxial strain and hydrostatic strain. Finally,we have calculated the phase diagram of SrTiO3 under uniaxial strain, as obtained fromLandau theory and discussed how it compares with the first-principles calculations.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / Neste trabalho, os efeitos de um campo magnético oblíquo externo no modelo de Gross- Neveu (2+1)-dimensional, que inclui as componentes paralela e perpendicular do campo em relação ao sistema, são estudados no contexto da simetria quiral e discreta do modelo. Nosso principal interesse está nos efeitos deste campo sobre o diagrama de fase do sistema, onde também incluímos os efeitos combinados de temperatura e potencial químico. Os diagramas de fase são obtidos através do potencial efetivo a 1 loop para o modelo, derivado em primeira ordem na expansão 1=N. Transições de fase relevantes que podem ser estudadas através deste modelo são, por exemplo, metal-isolante em matéria condensada e na teoria quântica de campos de férmions planares em geral. A relação entre a transição de fase com quebra da simetria quiral e discreta e o surgimento de um gap (ou a presença de um valor esperado no vácuo do campo escalar diferente de zero), como função do campo magnético oblíquo, é analisada em detalhes. / In this work, the effects of an external oblique magnetic field in the (2+1)-dimensional Gross-Neveu model, and that therefore includes both parallel and perpendicular components of the applied field, are studied in the context of the models discrete chiral symmetry. Our main concern is in the effects of such a field in the systems phase diagram and that also includes the combined effects of temperature and chemical potential. The phase diagrams are obtained through the one-loop effective potential for the model, derived in the leading order in the 1=N expansion Relevant phase transitions that can be studied through this model are, for example, metal-insulator ones in condensed matter and in the quantum field theory of planar fermions in general. The relation between the phase transition with (discrete) chiral symmetry breaking and the emergence of a gap (or the presence of a chiral nonvanishing vacuum expectation value) in the planar fermionic system, as a function of the external oblique magnetic field, is analyzed in details.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho estudamos a dependência com o tamanho do sistema dos observáveis relacionados com a transição de fase de 1 ordem do rupo de calibre Z(2) em 4 dimensões. Foram realizadas simulações de Monte Carlo numa rede cúbica para diferentes valores da aresta, utilizando o método do Banho Térmico para sortear os elementos do grupo na rede. / In this work we study the dependence of the observable related to the size of the system to the phase transition of first order of the Z(2) gauge group in 4 dimensions. Monte Carlo simulations were performed on a cubic lattice for different values of the edge, using the method of the Heat Bath to randomize the group members in the lattice.

Utilizando a técnica da matriz de transferência e as ideias de finite-size scaling, de renormalização fenomenológica e de invariância conforme estudamos dois modelos de polímeros interagentes na rede quadrada. Em ambos, a atividade de um monômero pertencente ao polímero vale x = e. Quando as interações são entre as ligações primeiras-vizinhas que pertencem ao polímero definimos um fator de Boltzmann associado à interação como sendo = e-l, onde l é a energia de interação entre ligações. Se as interações são entre os monômeros que pertencem a sítios primeiros-vizinhos mas não-consecutivos o fator de Boltzmann associado à interação é z = e-m , sendo m a energia de interação entre os monômeros. Através do estudo de pares de tiras de larguras L-L obtivemos estimativas para os diagramas de fases dos dois modelos. Ambos apresentam três fases: uma fase não-polimerizada, uma fase polimerizada usual e uma fase polimerizada densa, na qual o polímero se encontra colapsado. Nessa fase, o polímero assume uma configuração que maximiza o número de interações, de modo que a densidade de sítios ocupados pelo polímero tende a 1 e a densidade de interações assume valores muito próximos de 1. Os diagramas de fases dos dois modelos são qualitativamente semelhantes, havendo diferenças quantitativas, já esperadas. As transições entre a fase não polimerizada e a fase polimerizada densa são de primeira ordem. A transição entre a fase não polimerizada e a fase polimerizada usual é de segunda ordem, e um ponto dessa fronteira, que corresponde ao modelo sem interações ( = 1 ou z = 1, dependendo do modelo), é bem conhecido [1-3] e vale xc = 0,37905227 ± 0,00000012. A transição entre as fases polimerizadas é de primeira ordem para valores pequenos de x, mas muda para uma transição de segunda ordem quando x aumenta. Nessa fronteira existe um ponto tri crítico, que foi estimado em ( xPTC = 1,5 ± 0,1, yPTC = 1,1 ± 0,1 ), para o modelo de ligações interagentes. No caso do modelo de monômeros interagentes, não foi possível obter uma estimativa conclusiva a respeito da localização do ponto tri crítico. No encontro das três fronteiras existe um ponto crítico terminal, no qual terminam a linha de transições de segunda ordem entre a fase não polimerizada e a fase polimerizada usual, a linha de transições descontínuas entre as duas fases polimerizadas e a linha de transições de primeira ordem entre as fases não polimerizadas e a polimerizada densa. Os valores estimados por nós são (xPCT = 0,244 ± 0,002, yPCT = 3,86 ± 0,03) para o modelo de ligações interagentes e (xPCT = 0,345 ± 0,001, zPCT = 1,52 ± 0,001) para monômeros interagentes. O ponto , no qual termina a fronteira de segunda ordem entre a fase não-polimeriza e a fase polimerizada usual e onde ocorre pela primeira vez a transição de colapso é um ponto crítico terminal em ambos os modelos. Os expoentes críticos e associados à fronteira entre a fase não polimerizada e polimerizada usual também foram calculados, e encontramos os valores = 0,7507 ± 0,0008 e = 0,2082 ± 0,0004, para y = 1, e = 0,7498 ± 0,0004 e = 0,205 ± 0,003, para y = 1,2, para o modelo de ligações interagentes. Para o modelo de monômero interagentes, os dados foram = 0,7507 ± 0,0007 e = 0,2089 ± 0,0009, para z = 1, e = 0,7500 ± 0,0004 e = 0,205 ± 0,008, para z = 1,2. Observando os valores dos expoentes, vemos que eles ficam constantes dentro das barras de erros, de modo que a transição é uma transição de segunda ordem usual. Os valores concordam muito bem com os valores esperados, que são (exatamente) = 3/4 e = 5/24 [4]. / Using the transfer matrix technique, finite-size scaling, phenomenological renormalization group, and conformal invariance ideas, we studied the thermodynamic behavior of two interacting models of polymers on the square lattice. In both models one monomer that belongs to the polymer has an activity x = e. When the interactions are between first neighbor bonds that belong to the polymer, we define a Boltzmann factor y = e-l, where l is the interaction energy between two bonds. If the interactions are between monomers located at first neighbor but nonconsecutive sites, the associated Boltzmann factor is z = e-m, where m is the interaction energy between two monomers. We consider pairs of strips of widths L-L\' and found estimates for the phase diagrams of both models. They have three phases: a non-polymerized phase, an usual polymerized phase and a dense phase, in which the polymer is colapsed. In this phase, the configuration of the polymer is that maximizes the number of interactions, and the density of sites occupied by the polymer goes to 1, while the density of interactions is very close to 1. The phase diagrams of two models are qualitatively similar, but. there are quantitative differences between them, as we already expected. The transition between non polymerized phase and dense phase is of first order. The transition between non-polymerized phase and usual polymerized phase is of second order, and one point of this frontier, which corresponds to the non-interacting model (y = 1 ou z = 1, depending on the specific model), is well known [1-3J and has the value xc = 0,37905227 ± 0,00000012. The transition between the two polymerized phases is of first order for small values of x, and it changes to a second order transition when x increases. At this frontier there is a tri critical point, and we found ( xTCP = 1,5 ± 0,1, yTCP = 1,1 ± 0,1 ) for the interacting bond model. It was not possible to obtain a conclusive estimation of the location of the tri critical point for the model of interacting monomers. At the point that all transition lines meet there is a critical endpoint, in which the second order transition line between non polymerized phase and usual polymerized phase, the first order transition line between polymerized phases and the first order transition line between non polymerized phase and dense polymerized phases finish. We found (xCEP = 0,244 ± 0,002, yCEP = 3,86 ± 0,03) for the interacting bond model and (xCEP = 0,345 ± 0,001, zCEP = 1,52 ± 0,001) for the interacting monomer model. The point is where ends the second order transition between non polymerized phase and usual polymerized phase and at this point the collapse transition happens at the first time. Then, in our models, the point is a critical endpoint. We also found the critical exponents and of the second order transition line between non polymerized phase and usual 1\'olymerizcd phase. The values we obtained are = 0,7507 ± 0,0008 and = 0,2082 ± 0,0004, for y = 1, and = 0,7498 ± 0,0004 and = 0,205 ± 0,003, for y = 1,2, to the model of interacting bonds. The interacting monomers model has = 0,7507 ± 0,0007 and = 0,2089 ± 0,0009, for z = 1, and = 0,7500 ± 0,0004 and = 0,205 ± 0,008, for z = 1,2. Looking at these results we can see that the exponents remain constant within error bars, thus the transition is a usual second order transition. Furthermore, these values are in a very good agreement with the expected values, which are = 3/4 and = 5/24 [4].

O modelo de Ising dipolar bidimensional inclui, além da interação ferromagnética entre os primeiros vizinhos, interações de longo alcance entre os momentos de dipolo magnético dos spins. A presença da interação dipolar muda completamente o sistema, apresentando um rico diagrama de fase, cujas características têm originado inúmeros estudos na literatura. Além disso, a possibilidade de explicar fenômenos observados em filmes magnéticos ultrafinos, os quais possuem diversas aplicações em àreas tecnológicas, também motiva o estudo deste modelo. O estado fundamental ferromagnético do modelo de Ising puro é alterado para uma série de fases do tipo faixas, as quais consistem em domínios ferromagnéticos de largura $h$ com magnetizações opostas. A largura das faixas depende da razao $\\delta$ das intensidades dos acoplamentos ferromagnético e dipolar. Através de simulações de Monte Carlo e técnicas de repesagem em histogramas múltiplos identificamos as temperaturas críticas de tamanho finito para as transições de fase quando $\\delta=2$, o que corresponde a $h=2$. Calculamos o calor específico e a susceptibilidade do parâmetro de ordem, no intervalo de temperaturas onde as transições são observadas, para diferentes tamanhos de rede. As técnicas de repesagem permitem-nos explorar e identificar máximos distintos nessas funções da temperatura e, desse modo, estimar as temperaturas críticas de tamanho finito com grande precisão. Apresentamos evidências numéricas da existência de uma fase nemática de Ising para tamanhos grandes de rede. Em nossas simulações, observamos esta fase para tamanhos de rede a partir de $L=48$. Para verificar o quanto a interação dipolar de longo alcance afeta as estimativas físicas, nós calculamos o tempo de autocorrelação integrado nas séries temporais da energia. Inferimos daí quão severo é o critical slowing down (decaimento lento crítico) para esse sistema próximo às transições de fase termodinâmicas. Os resultados obtidos utilizando um algoritmo de atualização local foram comparados com os resultados obtidos utilizando o algoritmo multicanônico. / Two-dimensional spin model with nearest-neighbor ferromagnetic interaction and long-range dipolar interactions exhibit a rich phase diagram, whose characteristics have been exploited by several studies in the recent literature. Furthermore, the possibility of explain observed phenomena in ultrathin magnetic films, which have many technological applications, also motivates the study of this model. The presence of dipolar interaction term changes the ferromagnetic ground state expected for the pure Ising model to a series of striped phases, which consist of ferromagnetic domains of width $h$ with opposite magnetization. The width of the stripes depends on the ratio $\\delta$ of the ferromagnetic and dipolar couplings. Monte Carlo simulations and reweighting multiple histograms techniques allow us to identify the finite-size critical temperatures of the phase transitions when $\\delta=2$, which corresponds to $h=2$. We calculate, for different lattice sizes, the specific heat and susceptibility of the order parameter around the transition temperatures by means of reweighting techniques. This allows us to identify in these observables, as functions of temperature, the distinct maxima and thereby to estimate the finite-size critical temperatures with high precision. We present numerical evidence of the existence of a Ising nematic phase for large lattice sizes. Our results show that simulations need to be performed for lattice sizes at least as large as $L=48$ to clearly observe the Ising nematic phase. To access how the long-range dipolar interaction may affect physical estimates we also evaluate the integrated autocorrelation time in energy time series. This allows us to infer how severe is the critical slowing down for this system with long-range interaction and nearby thermodynamic phase transitions. The results obtained using a local update algorithm are compared with results obtained using the multicanonical algorithm.

On a compris ces dernières années que certaines mesures d'intrications sont un outil efficace pour la compréhension et la caractérisation de phases nouvelles et exotiques de la matière, en particulier lorsque les méthodes traditionnelles basées sur l'identification d'un paramètre d'ordre sont insuffisantes. Cette thèse porte sur l'étude de quelques systèmes quantiques à basse dimension où un telle approche s'avère fructueuse. Parmi ces mesures, l'entropie d'intrication, définie via une bipartition du système quantique, est probablement la plus populaire, surtout à une dimension. Celle-ci est habituellement très difficile à calculer en dimension supérieure, mais nous montrons ici que le calcul se simplifie drastiquement pour une classe particulière de fonctions d'ondes, nommées d'après Rokhsar et Kivelson. L'entropie d'intrication peut en effet s'exprimer comme une entropie de Shannon relative à la distribution de probabilité générée par les composantes de la fonction d'onde du fondamental d'un autre système quantique, cette fois-ci unidimensionnel. Cette réduction dimensionnelle nous permet d'étudier l'entropie aussi bien par des méthodes numériques (fermions libres, diagonalisations exactes, ...) qu'analytiques (théories conformes). Nous argumentons aussi que cette approche permet d'accéder facilement à certaines caractéristiques subtiles et universelles d'une fonction d'onde donnée en général.Une autre partie de cette thèse est consacrée aux trempes quantiques locales dans des systèmes critiques unidimensionnels. Nous insisterons particulièrement sur une quantité appelée écho de Loschmidt, qui est le recouvrement entre la fonction d'onde avant la trempe et la fonction d'onde à temps t après la trempe. En exploitant la commensurabilité du spectre de la théorie conforme, nous montrons que l'évolution temporelle doit être périodique, et peut même être souvent obtenue analytiquement. Inspiré par ces résultats, nous étudions aussi la contribution de fréquence nulle à l'écho de Loschmidt après la trempe. Celle-ci s'exprime comme un simple produit scalaire -- que nous nommons fidélité bipartie -- et est une quantité intéressante en elle-même. Malgré sa simplicité, son comportement se trouve être très similaire à celui de l'entropie d'intrication. Pour un système critique unidimensionnel en particulier, notre fidélité décroît algébriquement avec la taille du système, un comportement rappelant la célèbre catastrophe d'Anderson. L'exposant est universel et relié à la charge centrale de la théorie conforme sous-jacente. / In recent years, it has been understood that entanglement measures can be useful tools for the understanding and characterization of new and exotic phases of matter, especially when the study of order parameters alone proves insufficient. This thesis is devoted to the study of a few low-dimensional quantum systems where this is the case. Among these measures, the entanglement entropy, defined through a bipartition of the quantum system, has been perhaps one of the most heavily studied, especially in one dimension. Such a quantity is usually very difficult to compute in dimension larger than one, but we show that for a particular class of wave functions, named after Rokhsar and Kivelson, the entanglement entropy of an infinite cylinder cut into two parts simplifies considerably. It can be expressed as the Shannon entropy of the probability distribution resulting from the ground-state wave function of a one-dimensional quantum system. This dimensional reduction allows for a detailed numerical study (free fermion, exact diagonalizations, \ldots) as well as an analytic treatment, using conformal field theory (CFT) techniques. We also argue that this approach can give an easy access to some refined universal features of a given wave function in general.Another part of this thesis deals with the study of local quantum quenches in one-dimensional critical systems. The emphasis is put on the Loschmidt echo, the overlap between the wave function before the quench and the wave function at time t after the quench. Because of the commensurability of the CFT spectrum, the time evolution turns out to be periodic, and can be obtained analytically in various cases. Inspired by these results, we also study the zero-frequency contribution to the Loschmidt echo after such a quench. It can be expressed as a simple overlap -- which we name bipartite fidelity -- and can be studied in its own right. We show that despite its simple definition, it mimics the behavior of the entanglement entropy very well. In particular when the one-dimensional system is critical, this fidelity decays algebraically with the system size, reminiscent of Anderson's celebrated orthogonality catastrophe. The exponent is universal and related to the central charge of the underlying CFT.

Through our whole work we study the XY model with all its entirety, a particular spin model where spins are confined in a plane. We try to bring out a good understanding of this model with all different types of phases and phase transition, it undergoes in critical situations. We conceive of these external conditions from very different physical models like High Tc Superconductor, Ultracold atoms in optical lattice which are in focus of recent research. Firstly we model high Tc Superconductors with very simple 2D XY model to get an idea about the diamagnetic response exhibited by these materials when kept in a external magnetic field. This modeling is reasonable because most of the physics of cuprate High Tc Superconductors are governed by their 2D copper oxide planes which insists us to consider 2D models. Later we shifted to a more realistic 3D anisotropic XY model , as the coupling between cuprates plane may have a considerable role in devising physics of those materials. We particularly focus on the 2D to 3D crossover effect on magnetisation showed by these models, with keeping an eye on how all these can be relate to the experimentally acquired magnetisation profile of High Tc Supercondutors. On the second project we investigate on the phase diagram of a fully frustrated 2-leg ladder Bose Hubbard model. After mapping it properly to a classical model, a bi-layer Fully Frustrated XY model on square lattice, we found that the frustration leads to the emergence of a new phase "Chiral Mott insulator(CMI)" sandwiched between "Chiral Superfluid(CSF)" and "regular Mott insulator(MI)" phase. We divide the whole report into four parts. The first chapter is basically contain-ing introductory part comprising the motivation. In the second chapter we discuss various types of phases and phase transitions of the 2D & 3D XY models. We try to address their critical behaviors. In the third chapter and onwards we consider our model in external magnetic field and observe magnetisation in these systems. Here we specially focus on 2D to 3D crossover effect on magtisation measurement. Lastly in the fourth chapter we bring out a correspondence of XY model with the 2 leg ladder fully frustrated Bose Hubbard Model. There we report the emergence of a new phase, Chiral Mott Insulator(CMI) due to frustration in system.

Les atomes froids dans les réseaux optiques permettent d'avoir un contrôle sans précédent des états a N-corps fortement corrélés. Pour cette raison, ils représentent un excellent outil pour l'implémentation d'un « simulateur quantique », utile pour réaliser de manière expérimentale de nombreux hamiltoniens de systèmes d'intérêt physique. En particulier, ils rendent possible la création de champs de jauge artificiels; ces derniers permettant d'accéder à la physique du magnétisme frustré. Dans ce travail, il s'agit de s'intéresser à la thermodynamique des atomes froids, en abordant ce sujet de manière théorique et numérique. A ce jour, le Monte Carlo quantique est la méthode la plus efficace dans ce domaine. Néanmoins, en raison de ce qu'on appelle le « problème du signe », elle ne peut s'appliquer qu'à une classe restreinte de systèmes, et dont par exemple les systèmes frustrés ne font pas partie. L'intérêt de cette thèse est de développer une nouvelle méthode approximée fondée sur une approche Monte Carlo. La première partie de cette thèse est consacrée à des considérations de nature théorique sur la structure spatiale des corrélations classiques et quantiques. Ces résultats nous permettent de développer, dans une deuxième partie, une approximation nommée « champ moyen quantique ». Celle-ci permet de proposer, dans une troisième partie, une méthode numérique qu'on appelle « Monte Carlo du champ auxiliaire » et qu'on applique à des cas d'intérêt physique, notamment au réseau triangulaire frustré. / Cold atoms in optical lattices offer unprecedented control over strongly correlatedmany-body states. For this reason they represent an excellent tool for the implementation ofa “quantum simulator”, which can be used to realize experimentally several Hamiltonians ofsystems of physical interest. In particular, they enable the engineering of artificial gaugefields, which gives access to the physics of frustrated magnetism. In this work, we study thethermodynamics of cold atoms both from a theoretical and a numerical point of view. Atpresent days, the most effective method used in this field is the quantum Monte Carlo. Butbecause of the so-called “sign problem” it can only be applied to a limited class of systems,which for example do not include frustrated systems. The interest of this thesis is to developof a new approximated method based on a Monte Carlo approach. The first part of this workis dedicated to theoretical considerations concerning the spatial structure of quantum andclassical correlations. These results permit to develop, in the second part, an approximationcalled quantum mean-field. This latter allows to propose, in the third part, a numericalmethod that we call “auxiliary-field Monte Carlo” and that we apply to some systems ofphysical interest, among which the frustrated triangular lattice.

Dans ce travail de thèse nous avons étudié la stabilité structurale à très haute pression de nanotubes de carbone et de nitrure de bore à la fois in situ et après cycle de pression. Nous essayons de cette manière une première approche pour déterminer le rôle de paramètres comme la composition (C or BN), nombre de parois ou diamètre dans la limite de stabilité de la structure des nanotubes.Les deux premiers chapitres de la thèse nous permettent de faire une introduction aux aspects fondamentaux relatifs aux propriétés des nanotubes de carbone, suivie d’une présentation des méthodes de synthèse ainsi que des techniques expérimentales utilisées dans cette thèse. Les trois chapitres suivants permettent de présenter l’évolution structurale des trois systèmes étudiés: a) Des nanotubes de carbone monoparois de faible diamètre enrichis en chiralité (6,5), b) nanotubes de carbone triple-parois, et c) des nanotubes de nitrure de bore à parois multiple. Les pressions maximales de ces études ont été de 80, 72 et 50 GPa respectivement. Le collapse radial de la structure et la stabilité tubulaire des nano-objets ont été au centre de nos recherches. En particulier, les nanotubes de carbone à simple parois de chiralité (6,5) peuvent être préservés jusqu’à 50 GPa, pression à la quelle a lieu une transformation irréversible. De leur côté, les nanotubes à 3 parois ont pu être détectés jusqu’à environ 60 GPa, présentant en suite une transformation irréversible à 72 GPa. Enfin, les nanotubes de nitrure de bore ont montré une plus faible stabilité mécanique face à leurs analogues carbonés. De plus ils présentent une évolution vers toute une variété de morphologies, parmi lesquelles certaines ont été observées pour la première fois dans ce travail de thèse / This thesis work focuses on the structural stability of well-characterized carbon and boron nitride nanotubes under very high pressures both including their in situ study as well as after the pressure cycle. We try to provide in this way a first approach to determine the role of parameters as composition (C or BN), number of walls or diameter on the limit stability of nanotube structures.In the two first chapters, we provide a basic description of the theoretical aspects related to carbon nanotubes, we address their main synthesis methods as well as the experimental techniques used in this thesis to study these systems. In the three following chapters, we describe the structural evolution of three systems i) low diameter (6,5) chirality enriched single wall nanotubes ii) triple-wall carbon nanotubes and iii) multiwall boron nitride nanotubes. The maximum pressure attained in these studies were of 80, 72 and 50 GPa respectively.Both the radial collapse of the structure and the mechanical stability of the tubular structure under very high pressure are addressed in the study. In particular, after their collapse, the low-diameter (6,5) single walled carbon nanotubes can be preserved up to 50 GPa and above this value the tubes undergo an irreversible structural transformation. On its side, the triple wall systems could be detected up to ~ 60 GPa but their transformed irreversibly at 72 GPa. Finally boron nitride tubes have a low mechanical stability when compared with their carbon counterparts. Under high pressures they present transformations at different pressures to a variety of structural morphologies, some of them having been detected for the first time in this work